This paper introduces the Morrey Banach spaces on spaces of homogeneous type. We establish the Rubio de Francia extrapolation theory for Morrey Banach spaces on spaces of homogeneous type. Consequently, we obtain the boundedness of the Calderón–Zygmund operators and the Fefferman–Stein inequalities on Morrey Banach spaces on spaces of homogeneous type. We apply the results obtained for Morrey Banach space to study the Morrey spaces with variable exponents on spaces of homogeneous type.
We observe that the diameter of small (in a locally uniform sense) balls in C^1,1 sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to C^0, the diameter is arbitrarily close to twice the radius. Both results hold independently of the bracket-generating condition.
The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. They are particularly useful in areas such as network analysis and geometry processing, as computation of a GW distance involves solving for registration between the objects which minimizes geometric distortion. Although GW distances have proven useful for various applications in the recent machine learning literature, it has been observed that they are inherently sensitive to outlier noise and cannot accommodate partial matching. This has been addressed by various constructions building on the GW framework; in this article, we focus specifically on a natural relaxation of the GW optimization problem, introduced by Chapel et al., which is aimed at addressing exactly these shortcomings. Our goal is to understand the theoretical properties of this relaxed optimization problem, from the viewpoint of metric geometry. While the relaxed problem fails to induce a metric, we derive precise characterizations of how it fails the axioms of non-degeneracy and triangle inequality. These observations lead us to define a novel family of distances, whose construction is inspired by the Prokhorov and Ky Fan distances, as well as by the recent work of Raghvendra et al. on robust versions of classical Wasserstein distance. We show that our new distances define true metrics, that they induce the same topology as the GW distances, and that they enjoy additional robustness to perturbations. These results provide a mathematically rigorous basis for using our robust partial GW distances in applications where outliers and partial matching are concerns.
We disprove the conjecture of Georgakopoulos and Papasoglu that a graph with no K-fat H minor is quasi-isometric to a graph with no H minor. On the other hand, we show that the following weakening holds: any graph with no K-fat H minor is quasi-isometric to a graph with no 3-fat H minor.
In this article, using the existence of infinite equidistant subsets of closed balls, we first characterize the injectivity of ultrametric spaces for finite ultrametric spaces. This method also gives characterizations of the Urysohn universal ultrametric spaces. As an application, we find that the operations of the Cartesian product and the hyperspaces preserve the structures of the Urysohn universal ultrametric spaces. Namely, let ( X , d ) \left(X,d) be the Urysohn universal ultrametric space. Then, we show that ( X × X , d × d ) \left(X\times X,d\times d) is isometric to ( X , d ) \left(X,d) , and show that the hyperspace consisting of all non-empty compact subsets of ( X , d ) \left(X,d) and symmetric products of ( X , d ) \left(X,d) are isometric to ( X , d ) \left(X,d) . We also establish that every complete ultrametric space injective for finite ultrametric space contains a subspace isometric to ( X , d ) \left(X,d) .
We study geometric properties of the family of p-parallel bodies of a convex body K with respect to a gauge body E. In particular, we investigate various regularity properties of their boundaries by means of their 0-extreme vectors, aiming for extensions of several results known for the p = 1 case. We also analyze decomposition properties of convex bodies via the p-sum of their p-inner parallel bodies. To this end, we introduce a new convex body associated with K, which is related to the p-parallel bodies and the p-sum of convex bodies, the p-form body, and examine its properties. The latter provides improvements of inequalities involving p-inner parallel bodies, their support functions and mixed volumes.
We present an overview of the scientific activity of Ermanno Lanconelli, to whom this volume is dedicated on the occasion of his birthday.
In this article, we study Liouville type theorems of fully nonlinear elliptic partial differential equations on the Heisenberg group and obtain some nonexistence results of positive solutions of Heisenberg Hessian equations (and inequalities), including Heisenberg Hessian quotient equations (and inequalities) and mixed Heisenberg Hessian quotient equations (and inequalities). The proofs are based on integration by parts and the Newton-Maclaurin inequality.
In this article, we study the convergence rate of the following Yamabe-type flow Rϕ(t)m=0inMand∂∂tg(t)=2(hϕ(t)m−Hϕ(t)m)g(t)∂∂tϕ(t)=m(Hϕ(t)m−hϕ(t)m)on∂M{R}_{\phi \left(t)}^{m}=0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}M\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\left\{\begin{array}{l}\frac{\partial }{\partial t}g\left(t)=2\left({h}_{\phi \left(t)}^{m}-{H}_{\phi \left(t)}^{m})g\left(t)\\ \frac{\partial }{\partial t}\phi \left(t)=m\left({H}_{\phi \left(t)}^{m}-{h}_{\phi \left(t)}^{m})\end{array}\right.\hspace{0.33em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial M on a smooth metric measure space with boundary (M,g(t),e−ϕ(t)dVg(t),e−ϕ(t)dAg(t),m)\left(M,g\left(t),{e}^{-\phi \left(t)}{\rm{d}}{V}_{g\left(t)},{e}^{-\phi \left(t)}{\rm{d}}{A}_{g\left(t)},m), where Rϕ(t)m{R}_{\phi \left(t)}^{m} is the weighted scalar curvature, Hϕ(t)m{H}_{\phi \left(t)}^{m} is the weighted mean curvature, and hϕ(t)m{h}_{\phi \left(t)}^{m} is the average of the weighted mean curvature.
The cost functions considered are c(x,y)=h(x−y)c\left(x,y)=h\left(x-y), where h∈C2(Rn)h\in {C}^{2}\left({{\mathbb{R}}}^{n}) is homogeneous of degree p≥2p\ge 2 with a positive definite Hessian in the unit sphere. We study multivalued monotone maps with respect to that cost and establish that they are single-valued almost everywhere. Further consequences are then deduced.
We study Bloom-type two-weight inequalities for commutators of the Hardy-Littlewood maximal function and sharp maximal function. Some necessary and sufficient conditions are given to characterize the two-weight inequalities for such commutators.
We establish a version of the Cauchy integral formula for holomorphic functions on a polidisc, with values in a complex Fréchet space X{\mathfrak{X}}. We prove a vector-valued analog to Weierstrass’s theorem on sequences of holomorphic functions fν∈O(Ω,X){f}_{\nu }\in {\mathcal{O}}\left(\Omega ,{\mathfrak{X}}) converging uniformly on compact subsets of Ω⊂Cn\Omega \subset {{\mathbb{C}}}^{n}. We obtain a Cauchy-Kovalevskaja-type theorem, i.e., prove existence of X{\mathfrak{X}}-valued Cω{C}^{\omega } solutions to the Cauchy problem P(x,D)u=fP\left(x,D)u=f on U⊂ΩU\subset \Omega and (∂ju/∂xnj)∣xn=0=φj({\partial }^{j}u/\partial {x}_{n}^{j}){| }_{{x}_{n}=0}={\varphi }_{j} on U0=U∩{xn=0}{U}_{0}=U\cap \{{x}_{n}=0\}, j∈{0,1,…,m−1}j\in \left\{0,1,\ldots ,m-1\right\}, where P(x,D)≡∑∣α∣≤maα(x)DαP\left(x,D)\equiv {\sum }_{| \alpha | \le m}{a}_{\alpha }\left(x){D}^{\alpha } and aα∈Cω(Ω){a}_{\alpha }\in {C}^{\omega }\left(\Omega ), f∈Cω(Ω,X)f\in {C}^{\omega }\left(\Omega ,{\mathfrak{X}}) and φj∈Cω(Ω0,X){\varphi }_{j}\in {C}^{\omega }\left({\Omega }_{0},{\mathfrak{X}}), with 0∈Ω⊂Rn0\in \Omega \subset {{\mathbb{R}}}^{n} open and Ω0=Ω∩{xn=0}{\Omega }_{0}=\Omega \cap \left\{{x}_{n}=0\right\}. The existence of an open set 0∈U⊂Ω0\in U\subset \Omega and of a solution u∈Cω(U,X)u\in {C}^{\omega }\left(U,{\mathfrak{X}}) to the Cauchy problem requires the structural condition a(0,…,0,m)(0)≠0{a}_{\left(0,\ldots ,0,m)}\left(0)\ne 0 and relies, as well as in the classical case of scalar valued solutions, on the Cauchy integral formula and on Weierstrass’ theorem for X{\mathfrak{X}}-valued holomorphic functions.
In this article, we revise Monti’s results on blow-ups of H-perimeter minimizing sets in Hn{{\mathbb{H}}}^{n}. Monti demonstrated that the Lipschitz approximation of the blow-up, after rescaling by the square root of the excess, converges to a limit function for n≥2n\ge 2. However, the partial differential equation he derived for this limit function φ\varphi through contact variation is incorrect. Instead, the limit function solves the following equation weakly∂∂y1Δ0φ=0,\frac{\partial }{\partial {y}_{1}}{\Delta }_{0}\varphi =0,
We explicitly construct parameter transformations between gradient flows in metric spaces, called curves of maximal slope, having different exponents when the associated function and the associated metric space satisfy a suitable convexity condition. These transformations induce the existence of canonical bijective maps between gradient flows. We also prove the regularizing effects of gradient flows. To establish these results, we work directly with gradient flows instead of using variational discrete approximations which are often used in the study of gradient flows.
We consider generic rank two distributions on five-dimensional nilmanifolds and show that the analytic torsion of their Rumin complex coincides with the Ray-Singer torsion.
In this paper we study quasiconformal curves which are a special case of quasiregular curves. Namely embeddings $\Omega\rightarrow\mathbb{R}^m$ from some domain $\Omega\subset\mathbb{R}^n$ to $\mathbb{R}^m$, where $n\leq m$, which belong in a suitable Sobolev class and satisfy a certain distortion inequality for some smooth, closed and non-vanishing $n$-form in $\mathbb{R}^m$. These mappings can be seen as quasiconformal mappings between $\Omega$ and $f(\Omega)$. We prove that a quasiconformal curve always satisfies the analytic definition of quasiconformal mappings and the lower half of the modulus inequality. Moreover, we give a sufficient condition for a quasiconformal curve to satisfy the metric definition of quasiconformal mappings. We also show that a quasiconformal map from $\Omega$ to $f(\Omega)\subset \mathbb{R}^m$ is a quasiconformal $\omega$ curve for some form $\omega$ under suitable assumptions. Finally, we show the same is true when we equip the target space $f(\Omega)$ with its intrinsic metric instead of the Euclidean one.
When dealing with certain mathematical problems, it is sometimes necessary to show that some function induces a metric on a certain space. When this function is not a well renowned example of a distance, one has to develop very particular arguments that appeal to the concrete expression of the function in order to do so. The main purpose of this work is to provide several sufficient results ensuring that a function of two variables induces a distance on the real line, as well as some necessary conditions, together with several examples that show the applicability of these results. In particular, we show how a hypothesis about the sign of the cross partial derivative of the candidate to distance is helpful for deriving such kind of results.
We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.
